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Linear Representation of Relational Operations. Kenneth A. Presting University of North Carolina at Chapel Hill. Relations on a Domain. Domain is an arbitrary set, Ω Relations are subsets of Ω n All examples used today take Ω n as ordered tuples of natural numbers,

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linear representation of relational operations

Linear Representation of Relational Operations

Kenneth A. Presting

University of North Carolina at Chapel Hill

relations on a domain
Relations on a Domain
  • Domain is an arbitrary set, Ω
  • Relations are subsets of Ωn
  • All examples used today take Ωn as ordered tuples of natural numbers,

Ωn = {(ai)1≤i≤n | ai  N }

  • All definitions and proofs today can extend to arbitrary domains, indexed by ordinals
graph of a relation
Graph of a Relation
  • We want to study relations extensionally, so we begin from the relation’s graph
  • The graph is the set of tuples, in the context of the n-dimensional space
  • n-ary relation → set of n-tuples
  • Examples:

x2 + y2 = p → points on a circle, in a plane

z = nx + my + b → points in a plane, in 3-space

hyperplanes and lines
Hyperplanes and Lines
  • Take an n-dimensional Cartesian product, Ωn, as an abstract coordinate space.
  • Then an n-1 dimensional subspace, Ωn-1, is an abstract hyperplane in Ωn.
  • For each point (a1,…,an-1) in the hyperplaneΩn-1, there is an abstract “perpendicular line,” Ω x {(a1,…,an-1)}

Graph, Hyperplane, Perpendicular Line, and Slice

slices of the graph
Slices of the Graph
  • Let F(x1,…,xn) be an n-ary relation
  • Let the plain symbol F denote its graph:

F = {(x1,…,xn)| F(x1,…,xn)}

  • Let a1,…,an-1 be n-1 elements of Ω
  • Then for each variable xi there is a set

Fxi|a1,…,an-1 = { ωΩ | F(a1,…,ai-1,ω,ai,…, an-1}

  • This set is the xi’s which satisfy F(…xi…) when all the other variables are fixed
the matrix of slices
The Matrix of Slices
  • Every n-ary relation defines n set-valued functions on n-1 variables:

Fxi(v1,…,vn-1) = { ωΩ | F(v1,…,vi-1,ω,vi,…,vn-1) }

  • The n-tuple of these functions is called the “matrix of slices” of the relation F
properties of the matrix
Properties of the Matrix
  • Each slice is a subset of the domain
  • Each function Fxi(v1,…,vn-1) : Ωn-1 → 2Ω maps vectors over the domain to subsets of the domain
  • Application to measure theory
inverse map matrices to relations
Inverse Map: Matrices to Relations
  • Two-stage process, one step at a time
  • Union across columns in each row:

RowF(v1,…,vn-1) =

n | i<j → ai = vj

U { <ai>  Ωn | i=j → ai Fxj(v1,…,vn-1) }

j=1 | i>j → ai = vj-1

  • Union of n-tuples from every row:

F = U<vi>Ωn-1 RowF(v1,…,vn-1)

properties of the slicing maps
Properties of the Slicing Maps
  • Map from relations to matrices is injective but not surjective
  • Inverse map from matrices to relations is surjective but not injective
  • Not all matrices in pre-image of a relation follow it homomorphically in operations
boolean operations on matrices
Boolean Operations on Matrices
  • Matrices treated as vectors
  • i.e., Direct Product of Boolean algebras
    • Component-wise conjunction
    • Component-wise disjunction
    • Component-wise complementation
cylindrical algebra operations
Cylindrical Algebra Operations
  • Diagonal Elements
    • Images of diagonal relations, operate by logical conjunction with operand relation
  • Cylindrifications
    • Binding a variable with existential quantifier
  • Substitutions
    • Exchange of variables in relational expression
the diagonal relations
The Diagonal Relations
  • Matrix images of an identity relation, xi = xj
  • Example. In four dimensions, x2 = x3 maps to:
axioms for diagonals
Axioms for Diagonals
  • Universal Diagonal
    • dκκ = 1
  • Independence
    • κ {λ,μ} → cκ dλμ = dλμ
  • Complementation
    • κλ→ cκ (dκλ • F) • cκ (dκλ• ~F) = 0
cylindrical identity elements
Cylindrical Identity Elements
  • 1 is the matrix with all components Ω, i.e. the image of a universal relation such as xi=xi
  • 0 is the matrix with all components Ø, i.e. the image of the empty relation
diagonal operations are boolean
Diagonal Operations are Boolean
  • Boolean conjunction of relation matrix with diagonal relation matrix
  • Example
substitution is not boolean
Substitution is not Boolean
  • Substitution of variables permutes the slices – not a component-wise operation
  • Composition of Diagonal with Substitution

sκλF = cκ ( dκλ • F )

  • If we assume Boolean arithmetic, then standard matrix multiplication suffices
boolean matrix multiplication
Boolean Matrix Multiplication
  • Take union down rows, of intersections across columns
substitution operators
Substitution Operators
  • Square matrices, indexed by all variables in all relations
  • Substitution operator is the elementary matrix operator for exchange of columns
  • Example: in a four-dimensional CA, s32 =
axioms for cylindrification
Axioms for Cylindrification
  • Identity
    • cκ 0 = 0
  • Order
    • F + cκ F = cκ F
  • Semi-Distributive
    • cκ (F + cκ G) = cκ F + cκ G
  • Commutative
    • cκcλ F = cλcκ F
  • Take an n-ary relation, F = F(x1,…,xn)
  • Fix xi = a, that is, consider the n-1-ary relation F|xi=a = F(x1,…,xi-1,a,xi+1,…,xn)
  • Each column in the matrix of F|xi=a is:

Fxj|xi=a(v1,…,vn-2) =


cylindrification as union
Cylindrification as Union
  • Cylindrification affects all slices in every non-maximal column
  • Each slice in F|xi is a union of slices from instantiations:

Fxj|xi(v1,…,vn-2) = U Fxj|xi=a(v1,…,vn-2)


  • Component-wise operation
  • When cylindrification is defined as union of instantiations -
  • Matrix representations of relations form a cylindrical algebra.